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1. Khintchine-type double recurrence in abelian groups

   https://doi.org/10.1017/etds.2024.29  

   ACKELSBERG, ETHAN (April 2024, Ergodic Theory and Dynamical Systems)

   Abstract We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if$$\Gamma $$is a countable discrete abelian group,$$\varphi , \psi \in \mathrm {End}(\Gamma )$$, and$$\psi - \varphi $$is an injective endomorphism with finite index image, then for any ergodic measure-preserving$$\Gamma $$-system$$( X, {\mathcal {X}}, \mu , (T_g)_{g \in \Gamma } )$$, any measurable set$$A \in {\mathcal {X}}$$, and any$${\varepsilon }> 0$$, there is a syndetic set of$$g \in \Gamma$$such that$$\mu ( A \cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A ) > \mu(A)^3 - \varepsilon$$. This generalizes the main results of Ackelsberget al[Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107] and essentially answers a question left open in that paper [Question 1.12; Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107]. For the group$$\Gamma = {\mathbb {Z}}^d$$, the result applies to pairs of endomorphisms given by matrices whose difference is non-singular. The key ingredients in the proof are: (1) a recent result obtained jointly with Bergelson and Shalom [Khintchine-type recurrence for 3-point configurations.Forum Math. Sigma10(2022), Paper no. e107] that says that the relevant ergodic averages are controlled by a characteristic factor closely related to thequasi-affine(orConze–Lesigne) factor; (2) an extension trick to reduce to systems with well-behaved (with respect to$$\varphi $$and$$\psi $$) discrete spectrum; and (3) a description of Mackey groups associated to quasi-affine cocycles over rotational systems with well-behaved discrete spectrum. 

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2. Higher uniformity of arithmetic functions in short intervals I. All intervals

   https://doi.org/10.1017/fmp.2023.28  

   Matomäki, Kaisa; Shao, Xuancheng; Tao, Terence; Teräväinen, Joni (January 2023, Forum of Mathematics, Pi)

   Abstract We study higher uniformity properties of the Möbius function$$\mu $$, the von Mangoldt function$$\Lambda $$, and the divisor functions$$d_k$$on short intervals$$(X,X+H]$$with$$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$$for a fixed constant$$0 \leq \theta < 1$$and any$$\varepsilon>0$$. More precisely, letting$$\Lambda ^\sharp $$and$$d_k^\sharp $$be suitable approximants of$$\Lambda $$and$$d_k$$and$$\mu ^\sharp = 0$$, we show for instance that, for any nilsequence$$F(g(n)\Gamma )$$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when$$\theta = 5/8$$and$$f \in \{\Lambda , \mu , d_k\}$$or$$\theta = 1/3$$and$$f = d_2$$. As a consequence, we show that the short interval Gowers norms$$\|f-f^\sharp \|_{U^s(X,X+H]}$$are also asymptotically small for any fixedsfor these choices of$$f,\theta $$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$$L^2$$. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$$II$$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$$I_2$$sums. 

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3. Local parameters of supercuspidal representations

   https://doi.org/10.1017/fmp.2024.10  

   Gan, Wee Teck; Harris, Michael; Sawin, Will; Beuzart-Plessis, Raphaël (January 2024, Forum of Mathematics, Pi)

   Abstract For a connected reductive groupGover a nonarchimedean local fieldFof positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter$${\mathcal {L}}^{ss}(\pi )$$to each irreducible representation$$\pi $$. Our first result shows that the Genestier-Lafforgue parameter of a tempered$$\pi $$can be uniquely refined to a tempered L-parameter$${\mathcal {L}}(\pi )$$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of$${\mathcal {L}}^{ss}(\pi )$$for unramifiedGand supercuspidal$$\pi $$constructed by induction from an open compact (modulo center) subgroup. If$${\mathcal {L}}^{ss}(\pi )$$is pure in an appropriate sense, we show that$${\mathcal {L}}^{ss}(\pi )$$is ramified (unlessGis a torus). If the inducing subgroup is sufficiently small in a precise sense, we show$$\mathcal {L}^{ss}(\pi )$$is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is$${\mathbb {P}}^1$$and a simple application of Deligne’s Weil II. 

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4. On the ergodic theory of the real Rel foliation

   https://doi.org/10.1017/fmp.2024.6  

   Chaika, Jon; Weiss, Barak (January 2024, Forum of Mathematics, Pi)

   Abstract Let$${{\mathcal {H}}}$$be a stratum of translation surfaces with at least two singularities, let$$m_{{{\mathcal {H}}}}$$denote the Masur-Veech measure on$${{\mathcal {H}}}$$, and let$$Z_0$$be a flow on$$({{\mathcal {H}}}, m_{{{\mathcal {H}}}})$$obtained by integrating a Rel vector field. We prove that$$Z_0$$is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces$$({\mathcal L}, m_{{\mathcal L}})$$, where$${\mathcal L} \subset {{\mathcal {H}}}$$is an orbit-closure for the action of$$G = \operatorname {SL}_2({\mathbb {R}})$$(i.e., an affine invariant subvariety) and$$m_{{\mathcal L}}$$is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of$$Z_0$$with respect to any of the measures$$m_{{{\mathcal L}}}$$is zero. 

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5. Large monochromatic components in expansive hypergraphs

   https://doi.org/10.1017/S096354832400004X  

   Bal, Deepak; DeBiasio, Louis (March 2024, Combinatorics, Probability and Computing)

   Abstract A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary$$r$$-colouring of the complete$$k$$-uniform hypergraph$$K_n^k$$when$$k\geq 2$$and$$k\in \{r-1,r\}$$. We prove a result which says that if one replaces$$K_n^k$$in Gyárfás’ theorem by any ‘expansive’$$k$$-uniform hypergraph on$$n$$vertices (that is, a$$k$$-uniform hypergraph$$G$$on$$n$$vertices in which$$e(V_1, \ldots, V_k)\gt 0$$for all disjoint sets$$V_1, \ldots, V_k\subseteq V(G)$$with$$|V_i|\gt \alpha$$for all$$i\in [k]$$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on$$r$$and$$\alpha$$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms. Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary$$r$$-partite$$r$$-uniform hypergraph$$H$$with$$n$$edges in which every set of$$k$$edges has a common intersection. In this language, our result says that if one replaces the condition that every set of$$k$$edges has a common intersection with the condition that for every collection of$$k$$disjoint sets$$E_1, \ldots, E_k\subseteq E(H)$$with$$|E_i|\gt \alpha$$, there exists$$(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$$such that$$e_1\cap \cdots \cap e_k\neq \emptyset$$, then the smallest possible maximum degree of$$H$$is essentially the same (within a small error term depending on$$r$$and$$\alpha$$). We prove our results in this dual setting. 

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